This section is a mild generalization of the coalgebraic characterization of
weighted automata and weighted bisimulation in Bonchi et al. [1]} from the setting of vector spaces to semimodules.
We start by fixing the notation and recalling the basic notions of semimodules
and coalgebras. Then we show how weighted automata over a semiring \(K\)  can be
seen as a coalgebra over a functor \(\mathcal {W}: \mathsf {Set} \rightarrow \mathsf {Set}\)  and
characterize weighted bisimilarity \(\sim _w\)  as the behavioural equivalence
\(\approx _\mathcal {W}\)  for the functor \(\mathcal {W}\) . The results and proofs are essentially
identical to those in [1]}, as they do not use the
additional properties satisfied by vector spaces. We include them for the sake
of completeness. The only exception to this is the correspondence between
\(K\) -weighted bisimulation and kernels of \(\mathcal {W}\) -homomorphisms in
Section REF , which uses our definition of kernel and is therefore
new.
